G • COOLING BY PROTECTIVE FLUID FILMS 



where ul = —{R/2p)(dp/dx) has the dimension of the square of a ve- 

 locity. The closed form solution for velocity distribution in a fully de- 

 veloped turbulent pipe flow with fluid injection at the wall is obtained 

 from Eq. 5-55 with boundary conditions given in Eq. 5-50. This is 



K 



lni^J-2f 

 1 - f 



1 v^ 



In 



1 + r 

 1 -f 



2r 



(5-56) 



where f = (r/R)^, u^ is the velocity at the center of the pipe, and K is 

 an empirical constant to be discussed later. For a zero injection velocity 

 Eq. 5-56 reduces to the form expressing the velocity distribution in flow 

 through a circular pipe [36, pp. 340-344]. 



0.2 



0.4 



R 



0.6 



0.8 



1.0 



0.2 



0.4 



0.6 



u/ui 



0.8 



1.0 



1.2 



Fig. G,5f. Effect of fluid injection on the velocity distribution 

 in turbulent pipe flow. (From [35].) 



From Eq. 5-56 the mean flow velocity Wm can be determined by inte- 

 gration over the cross section. It is 



R'' Jo 



Um = ^F^ I urdr = u, — 



16 Up 

 TEH 



+ 0.541 



K' 



(5-57) 



Hence the empirical constant K can be readily calculated from Eq. 5-57 

 if the measured velocity distribution and pressure gradient are known. 

 The average value of K is about 0.24 for the case of zero injection. How- 

 ever, the values of K tend to increase with the increase of fluid injection. 

 Additional data from further tests (now being conducted at the Poly- 

 technic Institute of Brooklyn) are needed in order to form a definite 

 relationship between the above two parameters. In Fig. G,5f the effect 



( 472 > 



